CHAP. XX] BINARY QUADRATIC FORM MADE A CUBE. 537 



two cases z can be even without x and y being even. That the solution of 

 the equation is effected by different formulas according as z is even or odd 

 is shown by the case c = 47. Then all relatively prime solutions in which 

 z is odd are 



where / and g are relatively prime and one even. All solutions of 

 x +47?/ 2 = (2w) 3 , where u is odd, are 



x = 13/ 3 +60/ 2 0- 1GS/0 2 - 144f/ 3 , 



with similar expressions when z = 4u, Su, 16u, etc. The cases c = 35, 

 c = 499 are treated (p. 142, p. 155) 



Pepin 34 noted that 2x 3 = 3y 2 -l has the solution a; = 61, ?/ = 389, but left 

 undecided the question, of an infinitude of solutions. One of two methods 

 is based on the theorem that all relatively prime solutions of 2x 3 = 3y 2 z 2 

 are given by 

 z=/ 2 -3</ 2 , y=fA+3gB or 3fA-15gB, z=fA+3gB or -5fA+27gB, 



where A =/ 2 +9# 2 , B =/ 2 +<7 2 . It remains to find /, g such that z = 1. 



G. de Longchamps 35 stated that px 2 -\-qy 2 = z 3 always has integral solu- 

 tions. [In fact, a solution is x = at, y = &t, z = t=po?-\-qfP."^ 



H. Brocard 36 listed the known values of a for which x 3 y~ = a is impos- 

 sible and the values for which there is a single solution. 



E. B. Escott, 37 A. Cunningham and R. F. Davis 38 treated x 2 17 = y 3 . 



A. S. Werebrusow 39 expressed Euler's 6 solution of x 2 -{-cy 2 = z 3 in terms 

 of a=-2p, p=-p 2 +3cq 2 . 



Several 40 solved x 2 +3y 2 = 4z 3 completely by use of identities. 



U. Bini 41 gave a solution of x 2 -\-3y 2 = z 3 involving two parameters. 



An anonymous writer 42 noted that 17y 2 l = 2x 3 has no solution with 



A. Cunningham 43 gave a tentative method to solve x* = y 2 -\-a. Choose 

 a modulus m, preferably 10 3 or 10 4 , and find the values <m of x for which 

 x 3 a is a quadratic residue of m. By use of various m's we finally get 

 the possible linear forms of x. Application is made to a= 17, a= 127. 



Several 44 solved x 2 +xl = y 3 . 



Welsch 45 applied the theory of binary quadratic forms to justify 

 Legendre's 7 determination by use of V-3 of all solutions of x 2 +3y 2 = z 3 . 



34 Nouv. Ann. Math., (4), 3, 1903, 422-8. 



35 L'intermediaire des math., 9, 1902, 115. 



36 Ibid., 10, 1903, 284. 



37 Ibid., 12, 1905, 43-45. Amer. Math. Monthly, 26, 1919, 239-41. Cf. Brocard. 11 . 65 



38 Math. Quest. Educ. Times, (2), 8, 1905, 53-4. Cf. Cunningham. 43 



39 L'intermediaire des math., 10, 1903, 152. Cf. E. B. Escott, 11, 1904, 101-2. 



40 Ibid., 14, 1907, 168; 18, 1911, 279. 

 Ibid., 14, 1907, 192. 



42 Sphinx-Oedipe, 1906-7, 79. 



43 Math. Quest. Educ. Times, (2), 14, 1908, 106-8. 



44 L'intermediaire des math., 15, 1908, 244; 16, 1909, 201; 17, 1910, 126; 23, 1916, 4. 

 46 Ibid., 17, 1910, 179-180. 



